Recently, the notion of higher-order topology has been extended to the Dirac semimetals
(DSMs)[1]}, [2]}, [3]} as well as WSMs[4]}, [5]}. A Dirac point in three dimensions can
be understood as two degenerate Weyl nodes with opposite monopole charges, where crystalline symmetries
forbid the two Weyl nodes from hybridizing and opening a gap at each Dirac point[6]}. Given this
picture of the bulk, a three-dimensional (3D) DSM has two copies of arc-like surface states which
resemble Fermi-arcs in WSMs, i.e., the double Fermi arcs. However, unlike the surface states in the WSM,
the double Fermi arcs on the surface of a DSM are not topologically protected. They can be continuously
deformed into a closed Fermi contour without any symmetry breaking or bulk phase transition[7]}.
Therefore, they are not topological consequences of the bulk Dirac points themselves. In contrast, the
higher-order DSMs (HODSMs) can exhibit robust and nontrivial topology with spectroscopic consequences.
It has been proposed that the HODSM can be realized in the room- (\(\alpha \) ) and intermediate-temperature
(\(\alpha ^{\prime \prime }\) ) phases of Cd\(_3\) As\(_2\) , KMgBi, and rutile structure (\(\beta ^{\prime }\) -)
PiO\(_2\)[8]}, [9]}.
